Question

An apple loses \(4.5 \mathrm{kJ}\) of heat as it cools per \(^{\circ} \mathrm{C}\) drop in its temperature. The amount of heat loss from the apple per \(^{\circ} \mathrm{F}\) drop in its temperature is \((a) 1.25 \mathrm{kJ}\) \((b) 2.50 \mathrm{kJ}\) \((c) 5.0 \mathrm{kJ}\) \((d) 8.1 \mathrm{kJ}\) \((e) 4.1 \mathrm{kJ}\)

Short answer

Answer: (b) 2.50 kJ

Step-by-step solution

Identify the given heat loss per degree Celsius drop

The problem states that the apple loses 4.5 kJ of heat per degree Celsius drop in its temperature.

Use the conversion factor between Celsius and Fahrenheit

Since 1 degree Celsius change is equal to a 1.8-degrees Fahrenheit change, we can use this conversion factor to find out the heat loss per degree Fahrenheit.

Calculate the heat loss per degree Fahrenheit drop

To find the heat loss per degree Fahrenheit drop, we can divide the given heat loss per degree Celsius drop (4.5 kJ) by the conversion factor (1.8). That is, Heat loss per degree Fahrenheit drop = \(\frac{4.5 \mathrm{kJ}}{1.8} = 2.5\mathrm{kJ}\).

Match the result with the available options

Based on our calculation, the heat loss per degree Fahrenheit drop is 2.5 kJ, which matches option (b). So, the correct answer is \((b) 2.50\mathrm{kJ}\).

Key concepts

Temperature Conversion

Understanding temperature conversion is crucial when discussing thermal dynamics and heat energy. Temperature indicates the degree of heat within a substance and is measured using scales like Celsius (°C), Fahrenheit (°F), and Kelvin (K). These scales differ in their zero points and incremental values.

For example, the conversion between Celsius and Fahrenheit is a common task in thermodynamics exercises. Recognizing that the Celsius scale increments at a different pace than the Fahrenheit scale is vital. The conversion formula typically used is: \( F = \frac{9}{5}C + 32 \) for converting from Celsius to Fahrenheit and \( C = \frac{5}{9}(F - 32) \) for the reverse. Knowing how to switch between these scales allows us to relate temperature changes across different contexts and units.

In our apple example, understanding that a change of 1°C corresponds to a change of 1.8°F is key to determining heat loss in the units required. This conversion factor is essential when comparing thermal data from different regions of the world or from various scientific sources that may use different temperature units.

Heat Transfer

Heat transfer involves the movement of thermal energy from one object or substance to another due to a temperature difference. It's a fundamental concept in thermodynamics and it governs how energy is distributed in systems. There are three primary modes of heat transfer: conduction, convection, and radiation.

• Conduction is the transfer of heat through a solid material from molecule to molecule.
• Convection occurs in fluids (liquids and gases) where warmer areas of a fluid rise while cooler areas sink, creating a transfer of heat.
• Radiation is the transfer of heat through electromagnetic waves and does not require a medium.

In the case of the cooling apple, heat is likely being lost through convection to the surrounding air and possibly through radiation if the apple is warmer than the surrounding environment. It's important to note that heat always moves from a warmer object to a cooler one, striving for equilibrium within the environment.

Specific Heat Capacity

Specific heat capacity is a property that describes how much heat energy is needed to raise the temperature of a substance by one degree. It's a crucial factor in calculating the heat required to change a substance's temperature and is expressed in units of energy per mass per temperature change (e.g., J/kg°C).

Different substances have varying specific heat capacities, which means they will require different amounts of energy to achieve the same temperature change. For example, water has a high specific heat capacity, making it effective for temperature regulation.

In the exercise about the apple, the heat loss of 4.5 kJ per °C provides a direct application of this concept. This value, in essence, hints at the apple's specific heat capacity and allows us to understand how the apple's temperature will decline when losing a certain amount of heat. Combining knowledge of specific heat capacity with the exercise improvement advice, students can learn to calculate the temperature changes in various materials based on their specific heat capacities, which is a valuable skill in both academic and real-world settings.